The present disclosure relates to electronics and optics, and more specifically, to quantum computing.
In the field of quantum computing, computations are performed on a quantum state of the system, where the quantum system is viewed as being comprised of multiple particles. According to the postulates of quantum physics, any dynamical variable that can be measured in a system has a corresponding mathematical operator that acts on the system's wavefunction to “extract” the results of the measurement. Given such an operator, A1 (for a single particle #1) the postulate says that the measurement of the variable A corresponds to the mathematical action of operator A on the particle 1 component of the system's wavefunction, yielding for the observer the result of the measurement, a1,m, (the mth eigenvalue of dynamical variable A for particle 1) which assures the observer that after the measurement, the system is in the mth allowable eigenfunction |ΨM>Particle1 of A1 corresponding to the eigenvalue a1,m that was observed. Stated in mathematical terms:A1|ΨM>Particle1=a1,m|ΨM>Particle1 
Whenever there are two particles in a system, the system's wavefunction can be represented in a basis of operator A eigenstates (where A is position, momentum, energy, etc.) as a direct product:ΨM>Particle1|ΨN>Particle2,which yields different results for measurement of the dynamical variable A for particle 1 (A1 operator yields the mth eigenvalue a1,m) than the particle 2 (A2 operator yields the nth eigenvaluc a2,N):A1|ΨM>Particle1|ΨN>Particle2=a1,m|ΨM>Particle1|ΨN>Particle2, andA2|ΨM>Particle1|ΨN>Particle2=a2,n|ΨM>Particle1|ΨN>Particle2.
Prior to measurement, there is no reason why the state must be an eigenstate of any particular operator, because it is solely the act of measurement which defines the state. For example, it is possible for the state of the system prior to measurement to be in a linear superposition of state L and M for particle 1 as follows:
      [                                                                                  ψ                L                            ⁢                              >                                  Particle                  ⁢                                                                          ⁢                  1                                            +                                            ⁢                      ψ            M                          ⁢                  >                      Particle            ⁢                                                  ⁢            1                                      2              ]    ❘            ψ      N        ⁢          >              Particle        ⁢                                  ⁢        2            
The above state of the system is not regarded to be entangled—because the outcome of measurement of particle 2 is not coupled to the outcome of the measurement for particle 1. However, the state:
                                                                  ψ              1                        ⁢                          >                              Particle                ⁢                                                                  ⁢                1                                                              ⁢                  ψ          2                    ⁢              >                  Particle          ⁢                                          ⁢          2                    ⁢                        +                                                                ψ                2                            ⁢                              >                                  Particle                  ⁢                                                                          ⁢                  1                                                                                ⁢                  ψ          1                    ⁢              >                  Particle          ⁢                                          ⁢          2                            2        ,is entangled because if the result of measurement with A1 is 1 (which collapses the wavefunction into the first direct product in the above sum after measurement) then the outcome for particle 2 must be 2, and conversely if the outcome for particle 1 is 2 (yielding the second direct product in the above sum) then the outcome for particle 2 must be 1.
It can be shown from the time dependent Shrodinger equation, that if a single particle is in a pure eigenstate of the Hamiltonian, then this state is a standing wave whose spatial probability density (SPD) does not change in time. Conversely, when a single particle is in a linear superposition of two energy eigenstates, its shape changes and has a definite time evolution to it, and is not a standing wave. The two eigenstates in the superposition evolve with two different temporal phase terms so as to “beat against each other” in phase and the overall SPD oscillates with an angular frequency of oscillation proportional to the energy difference of the two states.